Isomorphisms between the Batalin-Vilkovisky antibracket and the Poisson bracket

TL;DR

This paper explores the mathematical relationships between different Lie algebras in Lagrangian field theories, establishing isomorphisms between the BRST cohomology, conserved currents, and charges, with implications for gauge theories.

Contribution

It proves isomorphisms between the BRST antibracket, Dickey bracket, and Poisson bracket in various gauge and non-gauge scenarios, clarifying their algebraic structures.

Findings

Subalgebra of BRST cohomology in ghost number -1 is isomorphic to conserved currents and charges in non-gauge theories.

In gauge theories, the algebra of conserved currents is isomorphic to a quotient of the BRST algebra.

An isomorphism between the antibracket and extended Poisson bracket is established in the Hamiltonian formalism.

Abstract

One may introduce at least three different Lie algebras in any Lagrangian field theory : (i) the Lie algebra of local BRST cohomology classes equipped with the odd Batalin-Vilkovisky antibracket, which has attracted considerable interest recently~; (ii) the Lie algebra of local conserved currents equipped with the Dickey bracket~; and (iii) the Lie algebra of conserved, integrated charges equipped with the Poisson bracket. We show in this paper that the subalgebra of (i) in ghost number $-1$ and the other two algebras are isomorphic for a field theory without gauge invariance. We also prove that, in the presence of a gauge freedom, (ii) is still isomorphic to the subalgebra of (i) in ghost number $-1$, while (iii) is isomorphic to the quotient of (ii) by the ideal of currents without charge. In ghost number different from $-1$, a more detailed analysis of the local BRST cohomology classes in the Hamiltonian formalism allows one to prove an isomorphism theorem between the antibracket and the extended Poisson bracket of Batalin, Fradkin and Vilkovisky.